Properties

Label 980.1.p.c
Level $980$
Weight $1$
Character orbit 980.p
Analytic conductor $0.489$
Analytic rank $0$
Dimension $8$
Projective image $D_{4}$
CM discriminant -4
Inner twists $16$

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Newspace parameters

Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 980.p (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.489083712380\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.137200.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -\zeta_{24}^{10} q^{2} -\zeta_{24}^{8} q^{4} -\zeta_{24}^{7} q^{5} -\zeta_{24}^{6} q^{8} + \zeta_{24}^{4} q^{9} +O(q^{10})\) \( q -\zeta_{24}^{10} q^{2} -\zeta_{24}^{8} q^{4} -\zeta_{24}^{7} q^{5} -\zeta_{24}^{6} q^{8} + \zeta_{24}^{4} q^{9} -\zeta_{24}^{5} q^{10} + ( \zeta_{24}^{3} + \zeta_{24}^{9} ) q^{13} -\zeta_{24}^{4} q^{16} + ( -\zeta_{24}^{5} + \zeta_{24}^{11} ) q^{17} + \zeta_{24}^{2} q^{18} -\zeta_{24}^{3} q^{20} -\zeta_{24}^{2} q^{25} + ( \zeta_{24} + \zeta_{24}^{7} ) q^{26} -\zeta_{24}^{2} q^{32} + ( -\zeta_{24}^{3} + \zeta_{24}^{9} ) q^{34} + q^{36} + 2 \zeta_{24}^{10} q^{37} -\zeta_{24} q^{40} + ( \zeta_{24}^{3} - \zeta_{24}^{9} ) q^{41} -\zeta_{24}^{11} q^{45} - q^{50} + ( \zeta_{24}^{5} - \zeta_{24}^{11} ) q^{52} + ( \zeta_{24} + \zeta_{24}^{7} ) q^{61} - q^{64} + ( \zeta_{24}^{4} - \zeta_{24}^{10} ) q^{65} + ( -\zeta_{24} + \zeta_{24}^{7} ) q^{68} -\zeta_{24}^{10} q^{72} + ( \zeta_{24}^{5} - \zeta_{24}^{11} ) q^{73} + 2 \zeta_{24}^{8} q^{74} + \zeta_{24}^{11} q^{80} + \zeta_{24}^{8} q^{81} + ( \zeta_{24} - \zeta_{24}^{7} ) q^{82} + ( -1 + \zeta_{24}^{6} ) q^{85} + ( -\zeta_{24} - \zeta_{24}^{7} ) q^{89} -\zeta_{24}^{9} q^{90} + ( -\zeta_{24}^{3} - \zeta_{24}^{9} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{4} + 4q^{9} + O(q^{10}) \) \( 8q + 4q^{4} + 4q^{9} - 4q^{16} + 8q^{36} - 8q^{50} - 8q^{64} + 4q^{65} - 8q^{74} - 4q^{81} - 8q^{85} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/980\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\) \(491\)
\(\chi(n)\) \(-\zeta_{24}^{4}\) \(-1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
−0.258819 0.965926i
0.258819 + 0.965926i
−0.965926 + 0.258819i
0.965926 0.258819i
−0.258819 + 0.965926i
0.258819 0.965926i
−0.965926 0.258819i
0.965926 + 0.258819i
−0.866025 0.500000i 0 0.500000 + 0.866025i −0.965926 + 0.258819i 0 0 1.00000i 0.500000 0.866025i 0.965926 + 0.258819i
79.2 −0.866025 0.500000i 0 0.500000 + 0.866025i 0.965926 0.258819i 0 0 1.00000i 0.500000 0.866025i −0.965926 0.258819i
79.3 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.258819 0.965926i 0 0 1.00000i 0.500000 0.866025i 0.258819 0.965926i
79.4 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.258819 + 0.965926i 0 0 1.00000i 0.500000 0.866025i −0.258819 + 0.965926i
459.1 −0.866025 + 0.500000i 0 0.500000 0.866025i −0.965926 0.258819i 0 0 1.00000i 0.500000 + 0.866025i 0.965926 0.258819i
459.2 −0.866025 + 0.500000i 0 0.500000 0.866025i 0.965926 + 0.258819i 0 0 1.00000i 0.500000 + 0.866025i −0.965926 + 0.258819i
459.3 0.866025 0.500000i 0 0.500000 0.866025i −0.258819 + 0.965926i 0 0 1.00000i 0.500000 + 0.866025i 0.258819 + 0.965926i
459.4 0.866025 0.500000i 0 0.500000 0.866025i 0.258819 0.965926i 0 0 1.00000i 0.500000 + 0.866025i −0.258819 0.965926i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 459.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
5.b even 2 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
20.d odd 2 1 inner
28.d even 2 1 inner
28.f even 6 1 inner
28.g odd 6 1 inner
35.c odd 2 1 inner
35.i odd 6 1 inner
35.j even 6 1 inner
140.c even 2 1 inner
140.p odd 6 1 inner
140.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.1.p.c 8
4.b odd 2 1 CM 980.1.p.c 8
5.b even 2 1 inner 980.1.p.c 8
7.b odd 2 1 inner 980.1.p.c 8
7.c even 3 1 980.1.f.e 4
7.c even 3 1 inner 980.1.p.c 8
7.d odd 6 1 980.1.f.e 4
7.d odd 6 1 inner 980.1.p.c 8
20.d odd 2 1 inner 980.1.p.c 8
28.d even 2 1 inner 980.1.p.c 8
28.f even 6 1 980.1.f.e 4
28.f even 6 1 inner 980.1.p.c 8
28.g odd 6 1 980.1.f.e 4
28.g odd 6 1 inner 980.1.p.c 8
35.c odd 2 1 inner 980.1.p.c 8
35.i odd 6 1 980.1.f.e 4
35.i odd 6 1 inner 980.1.p.c 8
35.j even 6 1 980.1.f.e 4
35.j even 6 1 inner 980.1.p.c 8
140.c even 2 1 inner 980.1.p.c 8
140.p odd 6 1 980.1.f.e 4
140.p odd 6 1 inner 980.1.p.c 8
140.s even 6 1 980.1.f.e 4
140.s even 6 1 inner 980.1.p.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.1.f.e 4 7.c even 3 1
980.1.f.e 4 7.d odd 6 1
980.1.f.e 4 28.f even 6 1
980.1.f.e 4 28.g odd 6 1
980.1.f.e 4 35.i odd 6 1
980.1.f.e 4 35.j even 6 1
980.1.f.e 4 140.p odd 6 1
980.1.f.e 4 140.s even 6 1
980.1.p.c 8 1.a even 1 1 trivial
980.1.p.c 8 4.b odd 2 1 CM
980.1.p.c 8 5.b even 2 1 inner
980.1.p.c 8 7.b odd 2 1 inner
980.1.p.c 8 7.c even 3 1 inner
980.1.p.c 8 7.d odd 6 1 inner
980.1.p.c 8 20.d odd 2 1 inner
980.1.p.c 8 28.d even 2 1 inner
980.1.p.c 8 28.f even 6 1 inner
980.1.p.c 8 28.g odd 6 1 inner
980.1.p.c 8 35.c odd 2 1 inner
980.1.p.c 8 35.i odd 6 1 inner
980.1.p.c 8 35.j even 6 1 inner
980.1.p.c 8 140.c even 2 1 inner
980.1.p.c 8 140.p odd 6 1 inner
980.1.p.c 8 140.s even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{1}^{\mathrm{new}}(980, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$3$ \( T^{8} \)
$5$ \( 1 - T^{4} + T^{8} \)
$7$ \( T^{8} \)
$11$ \( T^{8} \)
$13$ \( ( 2 + T^{2} )^{4} \)
$17$ \( ( 4 - 2 T^{2} + T^{4} )^{2} \)
$19$ \( T^{8} \)
$23$ \( T^{8} \)
$29$ \( T^{8} \)
$31$ \( T^{8} \)
$37$ \( ( 16 - 4 T^{2} + T^{4} )^{2} \)
$41$ \( ( -2 + T^{2} )^{4} \)
$43$ \( T^{8} \)
$47$ \( T^{8} \)
$53$ \( T^{8} \)
$59$ \( T^{8} \)
$61$ \( ( 4 + 2 T^{2} + T^{4} )^{2} \)
$67$ \( T^{8} \)
$71$ \( T^{8} \)
$73$ \( ( 4 - 2 T^{2} + T^{4} )^{2} \)
$79$ \( T^{8} \)
$83$ \( T^{8} \)
$89$ \( ( 4 + 2 T^{2} + T^{4} )^{2} \)
$97$ \( ( 2 + T^{2} )^{4} \)
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